Date: Aug 8, 2012 10:23 AM
Author: mina_world
Subject: Analysis with series 1/2^2+1/3^2

Hello teacher~

{(1/2^2) + (1/3^2) + (1/4^2) + ...}
+ {(1/2^3) + (1/3^3) + (1/4^3) + ...}
+ {(1/2^4) + (1/3^4) + (1/4^4) + ...}
+ ...

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I have a solution.(ambiguous)

Namely,

{(1/2^2) + (1/3^2) + (1/4^2) + ...}
+ {(1/2^3) + (1/3^3) + (1/4^3) + ...}
+ {(1/2^4) + (1/3^4) + (1/4^4) + ...}
+ ...

=

(1/2^2 + 1/2^3 + 1/2^4 + ...)
+ (1/3^2 + 1/3^3 + 1/3^4 + ...)
+ (1/4^2 + 1/4^3 + 1/4^4 + ...)

(this associative law ? Really possible?)

=

(1/2^2)/{1-(1/2)}
+ (1/3^2)/{1-(1/3)}
+ (1/4^2)/{1-(1/4)}
+...

= (1/2) + (1/3).(1/2) + (1/4).(1/3) + ...

= (1/2) + {(1/2)-(1/3)} + {(1/3)-(1/4)} + ...

= 1

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Hm... how do you think about it ?